This section provides background information related to the present disclosure which is not necessarily prior art.
One hot summer day on Aug. 14, 2003, with air conditioner demand for power high, the generating plant in Eastlake, Ohio, a suburb of Cleveland, went offline. This was at 1:31 pm. About thirty-minutes later a 345 kV high voltage line in a small town in Northeast Ohio, made contact with a tree, causing a short. This happened at 2:02 pm. Thereafter, at 3:05 pm another 345 kV line sagged into an overgrown tree in the town of Parma, south of Cleveland. All this activity caused a dip in the voltage on the Ohio portion of the electric power grid, setting into motion a cascade of further failures that, over the ensuing two hours, ultimately forced hundreds of power plants across New York, New Jersey, Maryland, Connecticut, Massachusetts, Michigan, Ohio and Ontario, Canada to shut down. By 4:13 pm 256 power plants were offline. 50 million people were without power. For many, the power outage lasted days.
The North American Electric Reliability Council, conducted an extensive investigation of the outage and detailed its findings in a report, “Technical Analysis of the Aug. 14, 2013, Blackout: What Happened, Why and What Did We Learn?,” Jul. 13, 20014. Instead of attributing the blackout to inanimate events, such as failure of an alarm processor (which had happened) or tree contact with a power line (which had happened), the Council found that the causes of the blackout were “rooted in deficiencies resulting from decisions, actions, and failure to act of the individuals, groups, and organizations involved.
Looking more deeply into why these individuals, groups, and organizations failed to properly react, the Council found, inter alia, that:                System operators did not have an effective contingency analysis capability for identifying contingency limit violations. Thus operators were essentially operating blind, with no way to effectively know which corrective actions would work and which would have worsened the problem.        System operators did not have effective diagnostic support. Critical real-time information about power grid conditions was not available to system operators, so they had no real-time knowledge by which to assess different contingency plans for corrective action. Again, operators were operating blind.        
This massive power outage did not happen instantaneously. Rather the outage was a cascade of events over the period of more than one hour. The following are some of those events:
13:31 EDT—Eastlake Unit 5 trips
14:02 EDT—Stuart-Atlanta 345-kV line trips
14:27 EDT—Star-South Canton 345-kV line trips and recloses
14:14 EDT—FirstEnergy control room alarm processor fails
14:20 EDT—FirstEnergy remote console fails
14:41 EDT—FirstEnergy primary server goes offline
15:05 EDT—Chamberlin-Harding 345-kV line trips, recloses, trips again and locks out
15:32 EDT—Hanna-Juniper 345-kV line trips
15:41 EDT—Star-South Canton 345-kV line trips and locks out
15:44 EDT—Babb-West Akron 138-kV line trips and locks out
15:45 EDT—Canton Central-Cloverdale 138-kV line trips and locks out [etc.]
The cascade began as illustrated above; it starting slowly but then quickly spread. The number of lines and generators lost stayed relatively low during the Ohio phase of the blackout, but then picked up speed after 16:08 EDT. The cascade was complete two-and-one-half minutes later. The outage spread from Ohio to Michigan and Ontario. From there it spread eastward along the Pennsylvania border to New York, where it proceeded to take out New Jersey and the eastern seaboard.
During the time the cascade event was happening, system operators might have taken corrective actions to contain the blackout to a local region, if only those operators could have seen a real-time picture of what was happening. Although system operators wanted to take corrective action to save the power grid, they lacked the ability to differentiate what corrective actions would make matters better and what would make matters worse. Thus, armed with the best technology available, these operators were forced to helplessly sit and watch the outage slowly spread and then built at frightening speed as Northeastern and Midwestern United States and Ontario Canada went dark.
In reality, power systems are currently operated close to their limits and transient stability analysis is one of the important tools in determining the operating limits. However, transient stability analysis (TSA) involves the extremely heavy computational burden of solving a large set of nonlinear differential equations. For example, to perform a TSA stability analysis of a 14,000-bus power system, the transient stability model would involve solving a set of some 15,000 differential equations and 40,000 non-linear algebraic equations for a time duration of 10-20 seconds.
An alternative to transient stability analysis involves so called direct methods. Direct methods can determine transient stability without the time-consuming numerical integration of a (post-fault) power system, however direct methods pose different problems arising from the fact that direct methods require that the region of attraction and the controlling unstable equilibrium point (UEP) must be ascertained. Both pose difficulties, since the boundary of the region of attraction (stability boundary) is not a smooth function but irregular, and the controlling UEP must be distinguished from other UEPs. Thus while direct methods of transient stability analysis have been successfully applied in evaluating the stability of power systems and deriving operating limits, the accuracy of these methods strongly relies on the determination of the controlling unstable equilibrium point (UEP). However, the numerical methods for the finding the correct controlling UEP frequently fail due to the presence of fractal shapes of the convergence region of the controlling UEP. To find the correct controlling UEP, using Newton methods for example, requires the—starting point of the search to be within a certain neighborhood of the desired solution. This requirement makes it difficult to find the best starting point and is computationally expensive.
The Newton method and its variations can converge if the initial starting point guess is within the region of convergence of the desired solution. In particular, in finding the correct UEP using traditional methods, the exit point and the minimum gradient point (MGP) have to be calculated first. The exit point is the point at which the projected fault-on trajectory exits the stability boundary of the post-fault stable equilibrium point (SEP). Computationally, the exit point is characterized by the first local maximum of the potential energy of the post-fault network along the projected fault-on trajectory.
Another technique using the Newton method involves detecting the exit point by detecting the change in the sign of the dot product of the post-fault power mismatch vector and the fault-on speed vector. The MGP is numerically characterized by the first local minimum value of the norm of the vector field of the post-fault trajectory. Most of the reported methods use the MGP as an initial point to generate a sequence of steps to find the controlling UEP. The robustness of finding the controlling UEP depends strongly on the accuracy of the calculation of MGP. An inaccuracy in detecting the exit point may cause difficulty in computing the MGP. Moreover, detecting an accurate exit point is computationally involved and sometimes requires the use of interpolation methods after bounding the exit point in a certain range. Thus, numerical inaccuracy in computing the exit point will likely cause failure of numerical methods to calculate the controlling UEP.
Several methods have been proposed to compute the controlling UEP including the Boundary of stability region based Controlling Unstable equilibrium point method (BCU method). However, due to the problems associated with detection of the exit point and consequently determination of the minimum gradient point, the BCU method may fail to find a solution. A continuation-based method that approximates the stability boundary locally and does not use the energy function to compute the controlling UEP was proposed in L. Chen, Y. Min, F. Xu, and K.-P. Wang, “A continuation-based method to compute the relevant unstable equilibrium points for power system transient stability analysis,” IEEE Trans. Power Syst., vol. 24, no. 1, pp. 165-172, February 2009. Notwithstanding the widespread use of the direct methods in transient stability analysis, the problem of precise determination of the controlling UEP and the speed of computation are still of concern of many researchers.
Another trend to compute the controlling UEP is to use homotopy-based approaches. Homotopy-based approaches have been successfully applied in computing the closest UEP. See, J. Lee and H.-D. Chiang, “A singular fixed-point homotopy method to locate the closest unstable equilibrium point for transient stability region estimate,” IEEE Trans. Circuits Syst. II, vol. 51, no. 4, pp. 185-189, April 2004; J. Lee, “A novel homotopy-based algorithm for the closest unstable equilibrium point method in nonlinear stability analysis,” Proceedings of the International Symposium on Circuits and Systems, ISCAS, vol. 3, pp. III-8-III-11, 2003.
The approaches reported in these papers utilize the strategy of singular fixed-point and the concept of bifurcation to locate the closest UEP. The algorithm proceeds by choosing a set of initial points that, by using homotopy-based approaches, converge to a set of type-1 UEPs. By choosing a set of proper initial points, the resulted set of the type-1 UEPs will include the closest UEP which can be distinguished by having lowest Lyapunov function value. However, closest UEP usually produces conservative results. Thus some have used the continuation-based methods without the use of the energy function and approximating the stability boundary locally. Again, this method gives very conservative results (not optimal) due to locally approximating the stability boundary.